5-dimensional equations and the initial-value problem

3.2 5-dimensional equations and the initial-value problem

The effective field equations are not a closed system. One needs to supplement them by 5D equations governing $ℰ μν$ , which are obtained from the 5D Einstein and Bianchi equations. This leads to the coupled system [281

]

1 £nK μν = K μαK αν − ℰμν − -Λ5g μν, (75 ) 6 £n ℰμν = ∇ αℬα (μν) + 1Λ5 (K μν − g μνK ) + K αβR μανβ + 3K α(μℰν)α − K ℰμν 6 + (KμαK νβ − K αβK μν)K α β, (76 ) β β £nℬ μνα = − 2∇ [μℰν]α + K α ℬμνβ − 2ℬ αβ[μK ν] , (77 ) £nR μναβ = − 2R μνγ[αK β]γ − ∇ μℬ αβν + ∇ μℬ βαν, (78 )

where the “magnetic” part of the bulk Weyl tensor, counterpart to the “electric” part $ℰμν$ , is

These equations are to be solved subject to the boundary conditions at the brane,

. ∇ μℰμν = κ45∇μ𝒮 μν, (80 ) . . ( 1 ) ℬμνα = 2∇ [μK ν]α = − κ25∇ [μ Tν]α − -gν]αT , (81 ) 3

where $A =.B$ denotes $A |brane = B|brane$ .

The above equations have been used to develop a covariant analysis of the weak field [281 ]. They can also be used to develop a Taylor expansion of the metric about the brane. In Gaussian normal coordinates, Equation (45 ), we have $£n = ∂∕∂y$ . Then we find

[ 1 ] gμν(x,y) = gμν(x,0) − κ25 Tμν + -(λ − T )gμν |y | 3 y=0+ [ 1 ( 2 ) 1 ( 1 ) ] + − ℰμν + --κ45 TμαT αν + --(λ − T )Tμν + -- -κ45(λ − T )2 − Λ5 gμν y2 + ... 4 3 6 6 y=0+ (82 )

In a non-covariant approach based on a specific form of the bulk metric in particular coordinates, the 5D Bianchi identities would be avoided and the equivalent problem would be one of solving the 5D field equations, subject to suitable 5D initial conditions and to the boundary conditions Equation (62

) on the metric. The advantage of the covariant splitting of the field equations and Bianchi identities along and normal to the brane is the clear insight that it gives into the interplay between the 4D and 5D gravitational fields. The disadvantage is that the splitting is not well suited to dynamical evolution of the equations. Evolution off the timelike brane in the spacelike normal direction does not in general constitute a well-defined initial value problem [8 ]. One needs to specify initial data on a 4D spacelike (or null) surface, with boundary conditions at the brane(s) ensuring a consistent evolution [147 , 242 ]. Clearly the evolution of the observed universe is dependent upon initial conditions which are inaccessible to brane-bound observers; this is simply another aspect of the fact that the brane dynamics is not determined by 4D but by 5D equations. The initial conditions on a 4D surface could arise from models for creation of the 5D universe [6 , 31 , 105

, 178 , 284 ], from dynamical attractor behaviour [247 ] or from suitable conditions (such as no incoming gravitational radiation) at the past Cauchy horizon if the bulk is asymptotically AdS.